
Here \(\mathcal {L}_{\text{EH}}\)  comes from the usual Einstein-Hilbert action, \(\mathcal {L}_{\Phi \Phi }\)  and \(\mathcal {L}_{\text{pot}}\)  are the kinetic and potential terms respectively of a bulk scalar field \(\hat{\Phi }(x,y)\) , \(\mathcal {L}_{\text{GHY}}\)  is the Gibbons-Hawking-York (GHY) boundary term [1]}, [2]}, and \(\Delta \mathcal {L}\)  is a useful total derivative we define in Appendix . The combination of \(\mathcal {L}_{\text{GHY}}\)  and \(\Delta \mathcal {L}\) 
is required to have a well-posed variational principle
for the gravitational action [3]}.
This Lagrangian is a function of the 5D metric \(G\) , which we parameterize in terms a 4D metric perturbation \(g_{\mu \nu }\)  and a scalar metric perturbation \(\hat{r}\)  as [4]}
\([G_{MN}] = \left(\hspace{-5.0pt}\begin{tabular}{ c c } {\,\,\,g_{\mu \nu } \,\, \exp {\left[-2\left(A(y)+\frac{ e^{2A(y)}}{2\sqrt{6}} \kappa \,\hat{r}(x,y)\right)\right]}} & {0} \\ {0} & {-\left(1 + \frac{ e^{2A(y)}}{\sqrt{6}} \kappa \,\hat{r}(x,y)\right)^{2}\,\,\,} \end{tabular}\hspace{-5.0pt}\right) \ .\) 
